= Prime ideal avoiding a multiplicative subset
In a <Noetherian ring>, an <ideal> $I$ disjoint from a nonempty <multiplicative subset> $S$ extends to an ideal $P$ maximal among those disjoint from $S$, by the <ascending chain condition>. It is proper. If $ab\in P$ but neither factor belongs to $P$, both $P+(a)$ and $P+(b)$ meet $S$. Multiplying such representatives puts an element of $S$ in $P$, a contradiction. Thus $P$ is a <prime ideal>. For a general <commutative ring> the same argument follows after applying the <Zorn lemma> to disjoint ideals; the union of a chain is disjoint from $S$.
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